C₅×#5731 #11756116

order	960
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#5731

Splittings

Fitting	C₃×C₅×C₄⋊Q₈ ⋊ C₂
Hall	C₃×C₅ ⋊ #541
Sylow	C₃ ⋊ C₅×#541

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×D₃	C₂³
hypercenter	C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#541
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊Q₈

384 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,3,4,5)(6,7,8)(9,11,12,15)(10,14,13,16)(18,24)(22,30)(23,32)(29,31);(10,13)(11,15)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,29,30,31);(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29);(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#80412, C₅×#80460, #14436245, …
maximal normal subgroup of	C₅×#80412, C₅×#80460, #14436245, …
maximal quotient of	C₅×#74358, C₅×#76483, C₅×#76611, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(960, 10311);